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Semantic facts on Kripke frames

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Abstract

This paper addresses the problem of how to represent semantic facts in possible worlds semantics. To this aim we associate a valuation function to every world in a Kripke frame that specifies the language of that world. The result is a two-dimensional semantics for which we present a complete axiomatization in a logical language that is based on standard modal logic. Lastly, we sketch possible philosophical applications of the framework.
Semantic Facts on Kripke Frames
Johannes Marti
Abstract
This paper addresses the problem of how to represent semantic
facts in possible worlds semantics. To this aim we associate a
valuation function to every world in a Kripke frame that speci-
fies the language of that world. The result is a two-dimensional
semantics for which we present a complete axiomatization in a
logical language that is based on standard modal logic. Lastly,
we sketch possible philosophical applications of the framework.
Keywords: two-dimensional modal logic, meaning change
1 Introduction
Many problems that bother philosophers of language crucially concern
the relation between semantic notions, such as for instance a sentence
expression pbeing true or pmeaning that ϕ, and modal or epistemic
notions, such as ϕbeing necessary or an agent knowing or believing
that ϕ. Widely known examples are Frege’s problem of informative
identity statements, the Twin Earth examples and semantic external-
ism, Kripke’s puzzle, the problem of radical interpretation, and the
epistemicist’s account of vague expressions. Nevertheless, there seems
to be no unified logical framework in which the interaction of semantic
and epistemic notions can be investigated.
The standard logical framework for modal or epistemic notions is
modal logic interpreted with possible worlds semantics. In modal logic
semantic questions are typically not considered. The modal formula
pis true, if pis true in all the worlds that are accessible from the
current world. It is not made explicit whether pis true at a world
because of the actual meaning of pand the non-semantic facts of the
epistemic alternatives, or because at that epistemic alternative the
meaning of the sentence expression pis such that it is true there.
2Johannes Marti
Usually it seems to be assumed that semantic facts are fixed and do
not vary across worlds.
In (Williamson, 1999) the modal logic B is used to deal with a
specific semantic problem. The universal modality ϕis interpreted
as “It is definite that ϕ and used to investigate higher-order vague-
ness. This interpretation of the modal operator implies that on the
semantic side worlds differ only with respect to the semantic facts but
not with respect to the atomic facts that language is about. This
approach is too restricted to a specific semantic notion to be useful
for us.
The formal system that is treated in this paper enriches possi-
ble worlds semantics with additional structure to explicitly represents
variation in semantic fact. The truth of an atomic expression at a
world is made relative to the language in which that expression is
evaluated. To access the semantic information contained in valuation
models we add a truth operator to the language of modal logic that
was already discussed in (Stalnaker, 1978). The resulting system can
be seen as a generalization of Stalnaker’s metasemantic interpretation
of two-dimensional semantics (Stalnaker, 2001, 2004).
The structure of this paper is as follows: First, in Section 2, we
introduce the semantic structures, we call them valuation models that
we use later. In Section 3 we discuss different notions of validity
in a well-suited modal language. In Section 4 the relation to two-
dimensional semantics is clarified. In Section 5 we sketch illustrative
applications of the framework to the problems of radical interpretation
and vagueness.
2 Valuation Models
The usual semantics for epistemic modal logic is based on Kripke
frames (Blackburn, Rijke, & Venema, 2002). Kripke frames are tuples
(W, R)where Wis any set and RW×Wis any relation on the set
W. The elements of Ware called worlds and the relation RW×W
accessibility relation.
In standard Kripke semantics one considers Kripke models which
are tuples (W, R, V ), where (W, R)is a Kripke frame and Vavalua-
tion, that is function V:Prop PWfor a set Prop of propositional
letters. One usually takes the valuation in a Kripke model to fix the
Semantic Facts on Kripke Frames 3
w1:
Vw1:¬p
Vw2:p
w2:
Vw1:p
Vw2:¬p
Figure 1: A Valuation Model
basic non-modal facts that are true at the worlds of the model. The
atomic fact pis true at all the worlds in the set V(p). In this paper we
take a different perspective on valuations. We think of every world
in a model as intrinsically containing information on which atomic
facts hold at that world. One can then take a valuation as assigning
meanings, that is the set of worlds where an expression is true, to
atomic sentences. On this view valuations are formal representations
of languages. Different valuations correspond to different ways how
language might be.
To represent semantic facts in Kripke frames, and hence semantic
knowledge for frames under the epistemic interpretation, we exploit
the fact that valuations are formal representations of languages. A
world in a Kripke frame should not only provide information about
its atomic and modal facts but it should also specify the semantic
facts that hold there. To do this we associate a different valuation
with every world of a frame. Formally, we define a valuation model
to be a tuple (W, R, Vw)wWwhere (W, R)is a Kripke frame and
Vw:Prop PWis a valuation for every wW, that is called the
valuation or language of the world w. The valuation Vsrepresents the
semantic facts that hold at the world w. In the situation corresponding
to the world wan atomic expression pProp has the meaning Vw(p).
The appropriate intuition to have for the languages in a valuation
model is that they are different ways how one natural language might
be. One should not think of them as being different natural languages
such as English or Dutch.
Figure 1 is an example of a valuation model. It contains the two
worlds w1where it is raining, and w2where the sun is shining. But
the worlds do not only differ in weather but also in semantic facts
because Vw1(p) = {w2}but Vw2(p) = {w1}. In the language of s1the
sentence pmeans that the sun is shining, hence pis true at w2but
4Johannes Marti
not at w1, whereas in the language of w2the sentence pmeans that
it is raining, hence pis true at w1but not at w2.
3 The Truth Operator
We now define the formal language LTwhich is based on standard
modal logic. Aside from the modal box modality it contains an oper-
ator Tto access the different valuations in a valuation models. The
precise syntax of the formal language LTis given by the grammar:
ϕ::= p|ϕϕ| ¬ϕ|ϕ|Tϕ.
where pProp is any propositional letter
The intended semantics for the T-operator is to switch the valua-
tion under which its subformula is evaluated to the valuation of the
current world. With our intuition that valuations are languages this
motivates calling Tthe truth operator for the language of the current
world. More precisely the truth conditions of LTare as follows: A for-
mula ϕ LTis satisfied at a world wWof a model (W, R, Vw)wW
under a valuation V:Prop PWif w, V |=ϕwhich is defined
inductively as:
w, V |=piff wV(p),
w, V |=ϕψiff w, V |=ϕand w, V |=ψ,
w, V |=¬ϕiff not w, V |=ϕ,
w, V |=ϕiff v, V |=ϕfor all vWwith Rwv,
w, V |= Tϕiff w, Vw|=ϕ.
If ϕdoes not contain any propositional letters that are not within the
scope of some truth operator its truth value does not depend on the
language Vand we can just say that ϕis true at wif w, V |=ϕfor
any language V.
There are different notions of validity that make sense in the con-
text of valuation models. Here we only consider validity and weak
completeness though it should be mentioned all definition and results
can be easily adapted to the more general notions of logical conse-
quence and strong completeness.
The most fundamental notion of validity results from varying the
world of evaluation and the language of evaluation independently. A
Semantic Facts on Kripke Frames 5
formula ϕ LTis a general validity if w, Vl|=ϕfor all worlds wand
lin any valuation model (W, R, Vw)wW.
Another notion of validity is obtained by requiring that the lan-
guage under which a formula is evaluated is the language of the world
in which the formula is evaluated. We define a formula ϕ LTto be
actual-language valid if w, Vw|=ϕfor all worlds win any valuation
model (W, R, Vw)wW.
An example of an actual-language validity that is not a general
validity is the equivalence schema Tϕϕ. It is an advantage of the
set of validities as opposed to the set of actual-language validities that
they are closed under -generalization, that is ϕis valid whenever
ϕis valid.
Two further notions of logical validity result from either fixing the
world of evaluation and quantifying over all languages or to fix the lan-
guage of evaluation and quantify over all worlds. We define a formula
ϕto be fixed-world valid if w0,Vw|=ϕfor all worlds win the valua-
tion model of the world w0. Similarly a formula ϕto be fixed-language
valid if w, Vw0|=ϕfor all worlds win the valuation model of the world
w0. The notions of fixed-world and fixed-language validity are inter-
esting because if we fix the world to be the actual world or we fix the
language to be actual English then these notions correspond roughly
to the notions of validity that stem from is called interpretational
and representational semantics in (Etchemendy, 1990). Special fixed-
world validities would include certain peculiar purely modal properties
of the actual world. In the context of epistemic logic this might be for
instance that a certain agent has inconsistent beliefs. Special fixed-
language validities could be certain atomic sentences that are analytic
truths such as for instance the English sentence “All bachelors are un-
married.”.
We only present an axiomatization of general validity since this
is much easier than actual language validity or even fixed-world or
fixed-language validity. The actual completeness proofs are rather
tedious and left out. The set of general validities in LTis a normal
modal logic in which and Tare both normal modal operators. A
complete axiomatization with respect to the class of valuation models
with arbitrary accessibility relations is given by a normal modal logic
K that is additionally closed under generalization for the T-operator
6Johannes Marti
and contains the following additional axiom schemata:
T(ϕψ)TϕTψTTϕTϕ
T¬ϕ ¬TϕTTϕTϕ
These axioms show how the truth operator distributes over the log-
ical connectives and operators. They correspond to our implicit as-
sumption that the logical symbols have the same meaning in all the
languages of a valuation model. For completeness with respect to
frames that have a transitive, reflexive relation or an equivalence re-
lation one can just add the S4 respectively S5 axiom schemata for the
-modality. If one considers transitive, Euclidean frames which are
not necessarily reflexive but possibly serial then it is not enough to
just add the usual K45 or KD45 axiom schemata for the -modality.
One can show that in these cases the additional axiom T(ϕϕ)
is needed to obtain completeness.
4 Two-Dimensional Semantics
Valuation models can be seen as a formalization of two-dimensional
semantics. The two-dimensional framework has been originally in-
troduced to give an account of the meaning for context-dependent
expressions such as indexicals and demonstratives (Kaplan, 1978).
Later, (Stalnaker, 1978) used a two-dimensional framework to model
how the meaning of an utterance can depend on the facts of the con-
text of utterance. The contextual factors that influence the meaning
of an utterance also include the possibility that the meaning of the
uttered expressions varies across contexts. With this interpretation of
two-dimensionalism, which has been called the “metasemantic” inter-
pretation, there is a close similarity between two-dimensional matrices
and valuation models, where the contexts in two-dimensional matrices
correspond to the languages in valuation models.
To explain the correspondence between valuation models and two-
dimensional matrices we consider again the example in Figure 1. The
situation in the valuation model of Figure 1 can also be represented
two-dimensionally by the following matrix:
p w1w2
Vw10 1
Vw21 0
Semantic Facts on Kripke Frames 7
In this matrix the columns correspond to worlds and the rows to the
languages of these worlds. As in two-dimensional semantics the box
operator quantifies universally along the horizontal of the matrix, be-
cause the original accessibility relation in Figure 1 is a total relation.
It is clear that for every valuation model with a total accessibility
relation we can find associated matrices for all the propositional let-
ters and whenever we have matrices for all propositional letters over
some fixed set of worlds we can find a corresponding valuation model
with total accessibility relation. This is the technical sense in which
valuation models are a generalization of two-dimensional matrices.
There is also a correspondence between the language LTand a
logical language suggested in (Stalnaker, 1978). We noticed in the
previous paragraph that the -modality has the same behavior in
both frameworks. In the matrix representation of valuation with to-
tal accessibility relation the T-operator shifts the valuation of formulas
along the vertical to the diagonal. This is exactly the semantics that
Stalnaker specifies for the dagger in (Stalnaker, 1978). Hence, the
two dimensional modal logic of and is exactly the modal logic with
truth operators where the underlying -modality is S5. In the case
where is a S5 modality LTalso corresponds to the fragment of the
operators and in the system B of (Segerberg, 1973), which is an
early study on two-dimensional modal logic. It shall be pointed out
that this logic is different from two-dimensional modal logics contain-
ing the modality and an actually operator @that has been studied
extensively for instance in (Gregory, 2001; Blackburn & Marx, 2002;
Stephanou, 2005). The actuality operators shifts along the horizontal,
the same dimension over which quantifies, to the diagonal whereas
the truth operator shifts along the vertical.
Given the similarity with two-dimensionalism one can apply the
framework of valuation models to the philosophical problems that
two-dimensional semantics has been used for. This concerns Frege’s
problem of informative identity statements, the Twin Earth examples
that have been used to argue for externalism of semantic content and
Kripke’s puzzle about a logically flawless individual that has appar-
ently inconsistent beliefs that it is aware of in different languages. In
(Stalnaker, 2004) it is sketched how these puzzles can be resolved with
a two-dimensional semantics under the metasemantic interpretation.
For a more detailed treatment of these examples in two-dimensional
semantics, although not under the metasemantic interpretation, we
8Johannes Marti
refer to (Chalmers, 2002).
5 Applications
In this section we shortly discuss possible applications of valuation
models and of modal logic with truth operators.
5.1 Radical Interpretation
In an epistemic logic with truth operators one can study the problem
of radical interpretation, as it is discussed in (Davidson, 1984) and
more formally in (Lewis, 1974). One might take radical interpretation
to be the task of constructing a valuation model to represent the
interpreted subject’s mental state from the evidence that is available
for interpretation. In this section we sketch how this might work in
a simple single agent case where it is assumed that the accessibility
relation is transitive and Euclidean and hence the logic of is K45.
We do not discuss any associated philosophical difficulties.
As the basic evidence for interpretation we take the sentences that
a subject assents to, or does not assent to, under various circum-
stances. Such evidence we express with sentences of the form “Af-
ter the subject has come to believe that ϕthen she does, or does
not, assent to ψ.”. Here ϕis a sentence in the metalanguage of the
interpreter that describes some change in the environment that the
subject observes, and ψis a sentence of the subject’s own language.
To express such sentences in the formal language LTwe have to find
formulas corresponding to the subject assenting to sentence ψand the
construction that something is true after the subject learned that ϕ.
We use the formula Tψto express formally that the subject as-
sents to the sentence ψ. So we require that the subject considers ψto
be true given what she believes about the meaning of the expressions
in ψ. The formulas of the form Tψcan be seen as a formal repre-
sentation of Davidson’s notion of the subject holding the sentence ψ
true in her own language. We take the assent to ψto express a belief
that Tψwhich in two-dimensional terminology is called a belief in the
diagonal proposition expressed by ψ. In (Stalnaker, 1978) it is argued
that an assertion of ϕnormally express the belief that ϕ, which is
called horizontal proposition in two-dimensional terminology. Only if
certain pragmatic principles are violated an assertion of ϕmight be
Semantic Facts on Kripke Frames 9
s0
s1:
Vs0:p
Vs1:¬a
Vs2:¬a
s2:
Vs0:¬p
Vs1:a
Vs2:a
Figure 2: The Subject’s Mental State
reinterpreted to express the diagonal Tϕ. In the context of radical
interpretation, however, it is justified to take all assertions to express
beliefs given by the formula Tϕinstead of just ϕbecause the sub-
ject’s own language might be very different from the interpreters own
metalanguage and it is the subject’s own language that is of interest.
The whole construction χholds after the subject learned that ϕ.”
can be modeled by formulas of the form [ϕ]χwhere [ϕ]is some sort
of update operator from dynamic epistemic logic (Baltag, Ditmarsch,
& Moss, 2008). Here we have not enough space to discuss how update
modalities can be made to work in a two-dimensional setting. For our
purposes it is enough to consider formulas of the shapes [ϕ]χand
[ϕ]¬χ, where χdoes not contain any -operators, to be equivalent
to (ϕχ)and ¬(ϕχ)respectively.
Now consider a simple example in which we observe that a subject
is assenting to the sentence aif it is not raining and assenting to ¬a
if it is raining. Moreover, the subject stays logically consistent so she
is not assenting to ¬aif it is not raining and not assenting to aif it is
raining. We can summarize this evidence with the following formulas
where we assume that pstands for the English metalanguage sentence
“It is raining.”:
[p]T¬a[¬p]Ta
[p]¬Ta[¬p]¬T¬a.
The possible mental states of the subject given the evidence can now
be taken to be all valuation models that satisfy the above formulas.
10 Johannes Marti
One such model is depicted in Figure 2. In this model the language
of the actual world s0is the metalanguage of the interpreter in which
the above formulas are evaluated. The propositional letters holding
at s0are not specified since the evidence for interpretation does not
constrain the atomic facts at the actual world. One might say that in
the model from Figure 2 ameans in the language of the subject that
it is not raining.
5.2 Vagueness
In the definition of valuation models of Section 2 languages are iden-
tified with valuations that assign a definite truth value, either true or
false, to every propositional letters at every world. One might wish
to loosen this requirement to accommodate for sentences that do not
clearly apply to a situation. An example of this phenomenon are sen-
tences containing vague expressions. In this Subsection I show how the
major frameworks that have been suggested to deal with the problem
of vague expressions (Williamson, 1994) lead to different ways how
one might loosen the requirement that every sentence in a language
is either true or false at a world.
One possible adaption to the notion of a language that allows for
sentences to be neither true nor false would be to allow the values
of propositional letters under a valuation to be at a world to be any
element from a set Uof generalized truth values. Valuation would then
have the type Prop UW, that is they are functions that assigns to
every propositional letter a function from states to to values in U.
In the case of standard classical valuations Uis just a two element
set. A more interesting example would be where Ucontains contain
three values: true, false, and undefined or all real numbers in the
interval [0,1]. There are two ways how one could accommodate for
many-valued valuations on the syntactic side. One possibility would
be to use new operators with the meaning of phas value r or “the
value of pis in A where rUand AUinstead of the usual
propositional letters. Alternatively one could also keep the standard
propositional letters and propagate the values Ualong the syntactic
tree to all formulas in the formal language LT. How this might be
done depends on the structure of U. In the three valued case one
might use the syntactic clauses of K3 or paraconsistent logic and use
ideas from (Fitting, 1992) for the modal operator.
Semantic Facts on Kripke Frames 11
Another way to obtain an account of semantic facts that does not
require every expression to be either true or false at a world would be
supervaluationism. In this case we identify a language of a world not
just with one valuation but with a set of valuations. Valuation models
are then structures of the form (W, R, Lw)wWwhere L(PW)Prop
is the set of valuation functions that is compatible with the semantic
facts holding at the world W. The satisfaction conditions of the truth
operator could then be changed to:
w, V |= Tϕiff w, U |=ϕfor all VLw.
This definition would give the truth operator the properties of super-
valuationistic super truth. It does for instance no longer distribute
over negations.
The valuation models from Section 2 with classic two-valued valu-
ations allow for an epistemicist’s solution to the problem of vagueness
if one lets the accessibility relation stand for epistemic uncertainty
and the modal for belief or knowledge. With this approach the
uncertainty that is involved in the use of vague expression is not at
the level of languages but is transferred to the epistemic uncertainty
that is modeled by the accessibility relation.
6 Conclusions and Future Work
In this paper we present valuation models as a straight-forward exten-
sion of Kripke models to represent semantic facts. We worked with
a simple modal language LTwhich has already been suggested in
two-dimensional semantics. This language is however not expressive
enough for our interpretation of the semantics. For instance one can
show that there is no formula in LTthat expresses that the proposi-
tional letter pmeans in the language of the current world that ϕ. It
is an aim for further work to investigate this problem and to extend
the formal language.
The relation between valuation models and other interpretations
of two-dimensional semantics should be investigated more thoroughly
from a philosophical perspective. Originally, two-dimensional seman-
tics has been a tool to capture context dependence and only under
the extreme metasemantic interpretation the context can also influ-
ence the language in which an expression is uttered. But exactly this
12 Johannes Marti
extreme case has interested us in this paper. It is an interesting ques-
tion how the more standard context dependence that arises in the
semantics of indexicals fits into the framework of valuation models.
In this paper we suggest applications of valuation models to the
problems of radical interpretation and vagueness but we do not discuss
them in any detail. We plan to investigate the application to radical
interpretation more thoroughly.
A further issue that is not addressed at all in this paper is how
different conceptions of what semantic facts are influence the formal
model. For instance, if semantic facts are just about the linguistic
disposition of agents it might be appropriate to assume epistemic in-
trospection. It might also be interesting to investigate how the choice
between internalistic and externalistic conceptions of semantic content
influences the framework and how this affects the contents of beliefs
that agents can have.
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Johannes Marti
ILLC, University of Amsterdam
P.O. Box 94242
1090 GE Amsterdam
e-mail: johannes.marti@gmail.com
URL: http://staff.science.uva.nl/~jfmarti/
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